Abstract

The Monte Carlo method of photon transport was used to simulate solar radiative transfer for cumulus-like cloud forms (and cloud fields) possessing structural characteristics similar to those induced by wind shear. Using regular infinite arrays of finite, slanted-cuboidal clouds (parallelepipeds), it was demonstrated that the magnitude of cloud field albedo variation as a function of relative solar azimuth angle (up to 40% of albedo) can be larger than the albedo disparities between plane-parallel clouds and fields of nonsheared finite clouds. In general, cloud field albedo is maximized when shearing is away from the sun and minimized when shearing is toward the sun. This is explained by changes in effective cloud fraction presented to the direct solar beam. The albedo of individual clouds, on the other hand, is maximized when shearing is toward the sun, especially when shearing angle equals solar zenith angle. This is because of both reduced irradiance onto cloud sides and enhanced effective optical depth of cloud. These results were corroborated by conducting similar experiments using realistic cloud forms generated by a dynamical/microphysical cloud model. The magnitude of albedo differences between sheared and corresponding nonsheared broken clouds reached 25% of the albedo. Again, this is due to differing effective cloud fractions and side illumination.

It was found that the bidirectional reflectance functions (BDRFs) of sheared clouds are sensitive to solar azimuth angle. Relative differences between BDRFs for clouds sheared toward and away from the sun can be as large as 50% for arrays of idealized parallelepiped clouds and 25% for more realistic clouds. Differences are minimized when viewing is perpendicular to the wind shear direction provided clouds are sheared toward or away from the sun. BDRFs for sheared clouds are much more asymmetric near the zenith than BDRFs for corresponding cubic (nonsheared) clouds. Hence, viewing sheared clouds at a 60° zenith angle will not necessarily provide least biased estimates of cloud field albedo as is the case for nonsheared clouds. Finally, it was demonstrated that BDRF differences arising from use of Mie and Henyey–Greenstein phase functions are substantially smaller than differences associated with varying solar azimuth angle.

Abstract

In an important paper written over 25 years ago, solar radiative fluxes based on exact Mie phase functions were compared to fluxes for corresponding elliptic and Henyey–Greenstein scattering phase functions. The poor performance of the elliptic function can be attributed to the method used to define its parameter. In this paper, a method is given that yields more appropriate elliptic phase functions. This greatly improves the credibility of the elliptic phase function as a potential candidate for use in two-stream approximations. This is especially true for aerosol conditions where it appears to be more suitable than the Henyey-Greenstein phase function.

Abstract

A method of computing grid-averaged solar radiative fluxes for horizontally inhomogeneous marine boundary layer cloud fields is presented. Its underlying assumptions are as follows: i) the independent pixel approximation (IPA) is applicable and ii) for regions the size of general circulation model (GCM) grid cells, frequency distributions of cloud optical depth τ can be approximated by gamma distribution functions. Equations are furnished for albedo and transmittance that, when applied to judiciously chosen spectral bands, require about three to four times as much CPU time as plane-parallel, homogeneous (PPH) two-stream approximations, which are ubiquitous to GCMs. This is not a hindrance, as two-stream solutions command typically less than 1% of a GCM's CPU consumption. This method, referred to as the gamma IPA, requires estimates of the mean and variance of τ for each applicable grid cell.

Biases associated with PPH models are assessed assuming that cloud properties in GCMs are tuned to yield albedos that agree with those inferred from satellite data. Thus, it is pertinent to ask: when cloud albedos for the gamma IPA and PPH models are forced to be equal, how do their cloud liquid water paths L, droplet effective radii r_e , and droplet absorptances differ? When albedos are equalized by altering ℒ (fixed r_e ), absorptance differences are generally within ±5%, but values of ℒ for the IPA exceed those for the PPH model often by much more than 20%, depending on ℒ and the extent of inhomogeneity. On the other hand, alteration of r_e , (fixed ℒ) requires that the IPA use smaller values of r_e than the PPR model. Therefore, since droplet single-scattering albedos increase with decreasing r_e , IPA absorptances are generally 5%–50% less than PPH absorptances, depending on ℒ and the extent of inhomogeneity. The overall implications are that by representing subgrid variability of marine boundary layer clouds in GCMs i) ℒ will increase, ii) r_e will decrease, and iii) there will probably he slightly less solar absorption by clouds relative to current values. Moreover, the magnitude of absorptance differences depend in part on the number of spectral bands J used to resolve the solar spectrum. In general, differences for J = 4 and J = 24 are approximately equivalent but for J<4, as in most GCMs, absorptance differences between the gamma IPA and PPH models are exaggerated and often of the wrong sign relative to those for J = 24.

Abstract

This study examines the ability to estimate regional cloud albedo using 1D series of cloud optical depth τ similar to those inferred from ground-based microwave radiometers. The investigation has two facets: use of appropriate radiative transfer algorithms and adequate portrayal of cloud structure. Using 1024 × 1024 pixel arrays of τ inferred from 28.5-m resolution Landsat data, regional albedos and albedos along individual scanlines are computed by a 3D Monte Carlo (MC) photon transport algorithm. Assuming the scanlines to be proxies for 1D series of τ a 2D MC algorithm and the Independent Pixel Approximation (IPA) are used to compute albedos for scanlines of various lengths and resolutions.

Regarding the appropriateness of doing radiative transfer calculations on a 1D series of τ, it is shown that for 1D series of τ containing 1024 pixels (∼30 km), lack of information about cloud structure adjacent to the series yields root-mean-square errors for 2D MC albedos of about 2% for a stratocumulus and 20% for two cumulus cloud fields. For the cumulus cases there is a marked tendency to over (under) estimate albedos for relatively bright (dark) scanlines. For the same series, the IPA performs very well relative to the, 2D MC for stratocumulus conditions. For broken cumulus clouds, however, notable biases between the 2D MC and IPA results stem mostly from the neglect of cloud sides by the IPA. In all cases, random errors are small.

The IPA is then used to investigate the accuracy of estimating regional cloud albedo with 1D datasets containing various amounts of information. It is demonstrated that for series with less than 100 pixels (≲3 km) at full resolution, the probability of attaining good estimates of regional cloud albedo is very low regardless of cloud type. Ideally, series with more than 1024 pixels should be used. Regarding sensitivity to data resolution, estimated regional albedos are almost resolution independent for pixel sizes up to about 2 km (∼70 pixels). At coarser resolutions. loss of cloud structure information important for radiative transfer is great and the quality of regional albedo estimates degraded, particularly for oblique sun.

Abstract

It has been hypothesized that over the past ∼200 years, industrial activity has enhanced the number of cloud condensation nuclei (CCN) in the lower atmosphere thereby reducing cloud droplet effective radii r _e and increasing the albedo of clouds. It is thought that in some regions, cloud albedos have increased so much that they have greatly ameliorated coincidental forcing by increased concentrations of greenhouse gases. The best estimates of this ameliorating effect come from large-scale climate/chemical transport models that assume clouds to be horizontally homogeneous at scales smaller than several hundred kilometers. It is demonstrated here that for a 2-μm reduction in r _e , conventional estimates of increased cloud albedo due to more CCN may be too large by up to, and possibly exceeding, 50%. The largest overestimates occur when reductions to r _e are accompanied by enhancements to both cloud variability and liquid water paths. This is attributed to fundamental differences in the way homogeneous and inhomogeneous clouds transport solar radiation.

Abstract

Solar radiative fluxes for broken, cumuloform cloud fields are examined from the point of view of subgrid parameterization for general circulation models (GCMs). A simple stochastic scaling model is used to simulate extensive broken cloud fields having horizontal variation of optical depth τ characterized by power-law wave-number spectra and power-law area/perimeter and cloud-size distribution properties. Fluxes are computed with the Monte Carlo method of photon transport.

Accurate flux estimates for extensive cloud fields are attainable with as few as 50 000 photons/simulation. Radiative fluxes for individual realizations of the cloud model represent the population well. Also, the effect of anisotropic scaling on azimuthally averaged fluxes may often be minimal. The latter two points are beneficial for flux parameterization purposes.

Solar fluxes for various scaling, regular, and plane-parallel broken cloud fields are compared. Scaling cloud fields the size of GCM grid boxes often produce significantly different reflectances from those produced by the extreme cases of plane-parallel and white noise arrays. When calculating fluxes at low sun periods, abundant small clouds should not be neglected. Reflectances for model cloud fields with horizontally variable τ are about 10%–15% smaller than those produced by the same cloud field but with all clouds having τ equal to the mean cloudy value of τ for the variable field. In some conditions, fluxes for extensive cloud fields are approximated well by both regular arrays and when horizontal transfer of photons is neglected.

Abstract

A statistical bidirectional method for including the effects of underlying reflecting surfaces in Monte Carlo simulations of atmospheric photon transport is presented. It is illustrated for the idealized Lambertian surface and a general particulate surface. It is shown that the simple Lambertian surface should be an adequate approximation (to mildly anisotropic surfaces) for calculation of overall fluxes for broken cloud fields. The problem of incorporating multiple internal reflections of solar radiation between cloud and surface is reviewed, and cloud field reflectances to upwelling radiation r_k as a function of the number of internal reflections k are calculated. Unlike plane-parallel and regular arrays of cloud, all r_k are generally not equal for more realistic scaling cloud fields. Strictly speaking, this prohibits use of the familiar geometric sum formulas for flux calculation in a multiple reflecting system. Fortunately, though fortuitously, the geometric sum formulas can be used in such cases if all r_k are assumed to equal approximately the spherical albedo of the cloud field.

Abstract

This paper examines the relative impacts on grid-averaged longwave flux transmittance (emittance) for marine boundary layer (MBL) cloud fields arising from horizontal variability of optical depth τ and cloud sides. First, using fields of Landsat-inferred τ and a Monte Carlo photon transport algorithm, it is demonstrated that mean all-sky transmittances for 3D variable MBL clouds can be computed accurately by the conventional method of linearly weighting clear and cloudy transmittances by their respective sky fractions. Then, the approximations of decoupling cloud and radiative properties and assuming independent columns are shown to be adequate for computation of mean flux transmittance.

Since real clouds have nonzero geometric thicknesses, cloud fractions  _c presented to isotropic beams usually exceed the more familiar vertically projected cloud fractions A _c . It is shown, however, that when A _c ≲ 0.9, biases for all-sky transmittance stemming from use of A _c as opposed to  _c are roughly 2–5 times smaller than, and opposite in sign to, biases due to neglect of horizontal variability of τ. By neglecting variable τ, all-sky transmittances are underestimated often by more than 0.1 for A _c near 0.75 and this translates into relative errors that can exceed 40% (corresponding errors for all-sky emittance are about 20% for most values of A _c ). Thus, priority should be given to development of general circulation model (GCM) parameterizations that account for the effects of horizontal variations in unresolved τ; effects of cloud sides are of secondary importance.

On this note, an efficient stochastic model for computing grid-averaged cloudy-sky flux transmittances is furnished that assumes that distributions of τ, for regions comparable in size to GCM grid cells, can be described adequately by gamma distribution functions. While the plane-parallel, homogeneous model underestimates cloud transmittance by about an order of magnitude when 3D variable cloud transmittances are ≲ 0.2 and by ∼20% to 100% otherwise, the stochastic model reduces these biases often by more than 80%.

Abstract

The two primary foci of this note are to assess the ability of the multilayer gamma-weighted two-stream approximation (GWTSA) to compute domain-averaged solar radiative fluxes and to demonstrate how its execution time can be reduced with negligible impact on performance. In addition to the usual parameters needed by a 1D solar code, the GWTSA requires ν ∈ R⁺, which depends on both the horizontal mean and mean logarithm of cloud water content. Reduced central processing unit (CPU) time is realized by simply rounding ν to the nearest whole number, denoted as [ν]. The experiment reported on here uses 120 fields generated by a 2D cloud-resolving model simulation of an evolving tropical mesoscale convective cloud system. Benchmark calculations are provided by the independent column approximation (ICA), and results are also shown for the conventional two-stream model.

The full GWTSA yields time- and domain-averaged broadband top-of-atmosphere albedo and surface absorptance values of 0.32 and 0.49, which are very close to the ICA values of 0.32 and 0.47. Correspondingly, the GWTSA using [ν] produces 0.34 and 0.46. In contrast, the conventional two-stream’s estimates are 0.56 and 0.20. While mean heating rate errors for the conventional two-stream average about −0.5 K day⁻¹ near the surface and almost +2 K day⁻¹ at 10 km, they are diminished at both altitudes to ∼0.25 K day⁻¹ for the GWTSA regardless of whether ν or [ν] is used. For this simulation, the GWTSA using [ν] requires just ∼25% more CPU time than the conventional two-stream approximation.

Abstract

A method for inferring cloud optical depth τ is introduced and assessed using simulated surface radiometric measurements produced by a Monte Carlo algorithm acting on fields of broken, single-layer, boundary layer clouds derived from Landsat imagery. The method utilizes a 1D radiative transfer model and time series of zenith radiances and irradiances measured at two wavelengths, λ ₁ and λ ₂, from a single site with surface albedos α _{λ 1} < α _{λ 2} . Assuming that clouds transport radiation in accordance with 1D theory and have spectrally invariant optical properties, inferred optical depths τ′ are obtained through cloud-base reflectances that are approximated by differencing spectral radiances and estimating upwelling fluxes at cloud base. When initialized with suitable values of α _{λ 1} , α _{λ 2} , and cloud-base altitude h, this method performs well at all solar zenith angles. Relative mean bias errors for τ′ are typically less than 5% for these cases. Relative variances for τ′ for given values of inherent τ are almost independent of inherent τ and are <50%. Errors due to neglect of net horizontal transport in clouds yield slight, but systematic, overestimates for τ ≲ 5 and underestimates for larger τ. Frequency distributions and power spectra for retrieved and inherent τ are often in excellent agreement. Estimates of τ depend weakly on errors in h, especially when h is overestimated. Also, they are almost insensitive to errors in surface albedo when α _{λ 1} is underestimated and α _{λ 2} overestimated. Reversing the sign of these errors leads to overestimation of τ, particularly large τ. In contrast, the conventional method of using only surface irradiance yields almost entirely invalid results when clouds are broken.

Though results are shown only for surfaces resembling green vegetation (i.e., α _{λ 1} ≪ α _{λ 2} ), the performance of this method depends little on the values of α _{λ 1} , and α _{λ 2} . Thus, if radiometric data have sufficient signal-to-noise ratios and suitable wavelengths can be found, this method should yield reliable estimates of τ for broken clouds above many surface types.